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Upper bounds of nodal sets for solutions of bi-Laplace equations: II

Jiuyi Zhu

Abstract

We investigate the upper bounds of nodal sets for solutions of bi-Laplace equations without using frequency functions which play an essential role in the study of nodal sets in the celebrated work by Logunov \cite{Lo18}. We obtain some delicate monotonicity and propagation of smallness results by Carleman estimates. A polynomial upper bound for the nodal sets of solutions is obtained.

Upper bounds of nodal sets for solutions of bi-Laplace equations: II

Abstract

We investigate the upper bounds of nodal sets for solutions of bi-Laplace equations without using frequency functions which play an essential role in the study of nodal sets in the celebrated work by Logunov \cite{Lo18}. We obtain some delicate monotonicity and propagation of smallness results by Carleman estimates. A polynomial upper bound for the nodal sets of solutions is obtained.
Paper Structure (4 sections, 10 theorems, 152 equations)

This paper contains 4 sections, 10 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. monotonicity of doubling index
  3. Upper bounds of nodal sets
  4. Appendix

Key Result

Theorem 1

Let $u$ be the solutions of bi-Laplace equations (bi-Laplace-1). There exists a positive constant $C$ that depends only on the manifold $\mathcal{M}$ such that where $\beta>\frac{1}{4}$ depends only on the dimension $n$.

Theorems & Definitions (20)

  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Remark 1
  • Lemma 3
  • Lemma 4
  • proof
  • Lemma 5
  • ...and 10 more